Texas Gov. Rick Perry accused the Obama administration Sunday of inaction on the flood of illegal immigrant children at the U.S. border with Mexico.

“This is a failure of diplomacy, it is a failure of leadership from the administration,” Perry said.

His interview on ABC’s “This Week” was one of several Sunday on television news shows that delved into the border crisis and the constraints that stand in the way of its resolution. Logistical and humanitarian issues, partisan quarreling and a Bush-era law are among the challenges the administration faces.

As Perry sees it, the key question is border security.

“What has to be addressed is the security of the border. You know that. I know that,” he said during the “This Week” interview. “The president of the United States knows that. I don’t believe he particularly cares whether or not the border of the United States is secure. And that’s the reason there’s been this lack of effort, this lack of focus, this lack of resources.”

Homeland Security Secretary Jeh Johnson, said Sunday the administration has dramatically sped up the processing of adults who enter the country illegally, and it is opening more detention facilities. He acknowledged on NBC’s “Meet the Press” that the unaccompanied children, some 9,700 taken into custody in May alone, pose the most vexing problem.

A George W. Bush-era law to address human trafficking prevents the government from returning the children to their home countries without taking them into custody and eventually through a deportation hearing. The administration says it wants more flexibility under the law.

President Barack Obama is asking Congress for more money and authority to send the children home, even as he also seeks ways to allow millions of other people already living in the U.S. illegally to stay.

A possible change to the law could give Border Patrol agents more leeway in handling these children.

The administration, Johnson said, is “looking at ways to create additional options for dealing with the children in particular, consistent with our laws and our values.”

Repeatedly pressed to say whether thousands of Central American children will be deported promptly, Johnson said, “We need to find more efficient, effective ways to turn this tide around generally, and we’ve already begun to do that.”

Johnson said the U.S. government is trying to send the message that all persons who enter the country illegally will face deportation proceedings eventually. In Central America, he said, “the criminal smuggling organizations are putting out a lot of disinformation about supposed free passes into this country” that will expire soon. “We’re cracking down on the smuggling organizations by surging law enforcement resources,” Johnson said.

Several Republicans and a Democrat said the administration has reacted too slowly and cautiously to the crisis.

More than 50,000 unaccompanied minors have been caught on the U.S.-Mexico border this year. Most are from Guatemala, Honduras and El Salvador, where a spike in violence and poverty are prompting parents to send their children on difficult and dangerous journeys north.

Unaccompanied Central American children generally are being released to relatives already in the United States. Mothers with their children often are released with a notice to appear later in immigration court.

Their numbers have overwhelmed federal agencies. When 140 would-be immigrants, mostly mothers with children, were flown to southern California to ease an overcrowded Texas facility, angry residents of Murrieta, Calif., greeted the bus as it pulled into town, complaining that they were being asked to do more than their share.

Alan Long, mayor of Murrieta, denounced the nation’s current immigration laws and practices on CNN’s “State of the Union,” Sunday. Central Americans think “they’re coming to a better place,” Long said, “but on that journey one-third of the females – some younger, in their teens – are raped.”

Rep. Henry Cuellar, D-Texas, also on “State of the Union,” said the administration “is one step behind” a major dilemma that was foreseeable. The number of children coming from Central America without adults has been rising for several years.

Sen. Lindsey Graham, R-S.C., in CBS’s “Face the Nation,” said that children entering the country illegally must be sent home.

If not, Graham said, “you’re going to incentivize people throughout that part of the world to keep sending their children here.”

Graham said foreign aid should be cut off to countries that don’t do more to discourage illegal immigration to the United States.